Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping Xianfeng Gu, Yaling Wang, Tony Chan, Paul Thompson, Shing-Tung Yau

Conformal Mapping Overview Map meshes onto simple geometric primitives Map genus zero surfaces onto spheres Conformal mappings preserve angles of the mapping Conformally map a brain scan onto a sphere

Example of Conformal Mapping

Overview Quick overview of conformal parameterization methods Harmonic Parameterization Optimizing using landmarks Spherical Harmonic Analysis Experimental results Conclusion

Conformal Parameterization Methods Harmonic Energy Minimization Cauchy-Riemann equation approximation Laplacian operator linearization Angle based method Circle packing

Cauchy-Riemann equation approximation Compute a quasi-conformal parameterization of topological disks Create a unique parameterization of surfaces Parameterization is invariant to similarity transformations, independent to resolution and it is orientation preserving

Cauchy-Riemann example

Laplacian operator linearization Use a method to compute a conformal mapping for genus zero surfaces by representing the Laplace-Beltrami operator as a linear system

Laplacian operator linearization

Angle based method Angle based flattening method, flattens a mesh to a 2D plane Minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensional space

Angle Based method example

Circle packing Classical analytical functions can be approximated using circle packing Does not consider geometry, only connectivity

Circle Packing example

Harmonic energy minimization Mesh is composed of thin rubber triangles Stretch them onto the target mesh Parameterize the mesh by minimizing harmonic energy of the embedding The result can be also used for harmonic analysis operations such as compression

Example of spherical mapping

Harmonic Parameterization Find a homeomorphism h between the two surfaces Deform h such that it minimizes the harmonic energy Ensure a unique mapping by adding constraints

Definitions K is the simplicial complex u,v are the vertices {u,v} is the edge connecting two vertices f, g represent the piecewise linear functions on K represents vector value functions represents the discrete Laplacian operator

Math overview Have a space Cpl, piecewise linear functions String constants kuv for each edge String energy is a piecewise linear function between the two vertices

Math II If kuv=1, tuette energy The parameters come out to be 1/2 cotangent of the angle Cotangent is minimized at 90 degrees, hence minimizing the harmonic energy will tend to conformal mappings

Math III #Harmonic energy is the sum of energies of the vector functions Only harmonic if no tangent and has a normal

Steepest Descent Algorithm T is the step size Steepest descent is going in the opposite direction of greatest gradient change

Conformal Spherical Mapping By using the steepest descent algorithm a conformal spherical mapping can be constructed The mapping constructed is not unique; it forms a Mobius group

Mobius group example Both a and c are conformal mappings of the face, but the locations of the poles are different, giving a different parameterization

Mobius group In order to uniquely parameterize the surface constraints must be added Use zero mass-center condition and landmarks In order to obtain a unique parameterization, more has to be done

Zero mass-center constraint The mapping satisfies the zero mass-center constraint only if All conformal mappings satisfying the zero mass-center constraint are unique up to the rotation group F arrow is the vector value piecewise linear function, sigma M1 is the area element on M1

Algorithm

Algorithm II

Algorithm IIb

Landmarks Landmarks are manually labeled on the brain as a set of uniformly parameterized sulcal curves The mesh is first conformally mapped onto a sphere An optimal Mobius transformation is calculated by minimizing Euclidean distances between corresponding landmarks

Landmark Matching Landmarks are discrete point sets, which mach one to one between the surfaces Landmark mismatch functional is Point sets must have equal number of points, one to one correspondence U is a member of omega, the group of mobius transformations

Landmark Example

Spherical Harmonic Analysis Once the brain surface is conformally mapped to , the surface can be represented as three spherical functions: This allows us to compress the geometry and create a rotation invariant shape descriptor

Geometry Compression Global geometric information is concentrated in the lower frequency components By using a low pass filter the major geometric features are kept, and the detail removed, lowering the amount of data to store

Geometry compression example

Shape descriptor The original geometric representation depends on the orientation A rotationally invariant shape descriptor can be computed by Only the first 30 degrees make a significant impact on the shape matching

Shape Descriptor Example

Experimental Results The brain models are constructed from 3D MRI scans (256x256x124) The actual surface is constructed by deforming a triangulated mesh onto the brain surface

Results By using their method the brain meshes can be reliably parameterized and mapped to similar orientations The parameterization is also conformal The conformal mappings are dependant on geometry, not the triangulation

Conformal parameterization of brain meshes

Different triangulation results

Results continued Their method is also robust enough to allow parameterization of meshes other than brains

Conclusion Presented a method to reliably parameterize a genus zero mesh Perform frequency based compression of the model Create a rotation invariant shape descriptor of the model

Conclusion continued Shape descriptor is rotationally invariant Can be normalized to be scale invariant 1D vector, fairly efficient to calculate The authors show it to be triangulation invariant Requires a connected mesh - no polygon soup or point models Requires manual labeling of landmarks

Questions?